Desperately Seeking Relevance: Music Theory Today [5]

Thus while it is commendable for composers to be concerned with the limitations of the senses, it is well to remember that music is directed, not to the senses, but through the senses and to the mind. And it might be well if more serious attention were paid to the capacity, behavior, and abilities of the human mind.

–Leonard B. Meyer,
Music, the Arts, and Ideas

▲

THE UBIQUITOUS TRIAD

At conception, roughly 500 years ago, the tonal triad – barely defined, almost invisible – was all potential, a gift waiting to be unwrapped.

Then came history. A lot of history. Today we've arrived at the end of that history.

Now, the triad-as-we-know-it-today is ubiquitous, fetishized, decoupled, anthropomorphized, overused, tired.

But most of all ubiquitous. This is the perfect word for it. It's not only that it is present everywhere, having invaded and pervaded the musics of virtually every culture on the planet. It's not only that its sound has captured the ears of most children even before they begin to talk. The concept of "ubiquity" originated as the Lutheran doctrine of the omnipresence of the body of Christ. The triad came to be heard as the omnipresent body of Music in the same mysterious sense. But ultimately came the whispering voice of the eternal devil lurking just outside the door of the Workshop, and of course the less subtle shout from the devil we invented:

An idea in music consists principally in the relation of tones to one another. But every relationship that has been used too often, no matter how extensively modified, must finally be regarded as exhausted; it ceases to have power to convey a thought worthy of expression.[1]

What chord is the robot playing, and why?
(Image from  CS4FN, Queen Mary, University of London)

As a fundamental compositional object, our triad has had a good run. But today its theory, now capable of speaking only through the many voices of analysis[2], has ossified into dogma. It's time to let it rest. Our composers–––not the gondoliers, but the explorers–––left it behind a century ago to map new coastlines and interiors. Now we all have to let go. But how?

.    .    .    .    .    .    .

Specific justifications for asserting the preeminence of the tonal triad, correctly but misleadingly referred to as the triad's multiple "natures," continue to multiply in the academic community. All of these natures/justifications taken together are considered by many analysts to be the basis for a demonstration of the inevitability of the tonal triad as the foundation for "our" music, coincidentally the music most amenable to extended analyses. It's as if there is a belief floating around out there that the more natures that theorists can identify or manufacture, the more solid a case can be made that the triad is a natural object whose status can't be challenged without bringing down the entire world of music. But there does come a day ––– "The lady doth protest too much, methinks." ––– when hollow echoes from that groaning tower of justifications makes us suspect that there's nothing left to justify (if there ever was a need). Except, perhaps, the justifications.

As we shall see, other sonorities (I will work out just one in the next post to serve as an example) may also have interesting and compositionally suggestive multiple natures –– some natures will be shared with the old workhorse, others will be different. First, to know what we will be looking for and to be able to contrast and compare any new music theory with the established, we need to briefly discuss what I consider to be the closest thing music theory today has to a set of axioms. Of course, that is not precisely what they are (even less are they Euclidean requests!) but they do share the axiomatic sense of being proposition sets–––rules, if you will–––some version of which, however warped, is required for any pitch-based music game. A complete list of quasi-axioms for the tonal triad would be unwieldy, but I believe all of them fall into four basic categories that (unintentionally?) mix facts and claims. Remember, these refer to the triad, not its most compatible matrix, the diatonic system.

1. Form inducing: The triad's structure invites characteristic compositional procedures and techniques such as "parsimonious" voice leading, modulation, chromaticization through decoupling from the diatonic, and so on.  It's in this category that some of tonal music theory's biggest claims and most bewildering terminological tangles are found. The triad's form inducing properties are arguably a sine qua non for tonal theories from Fux and Rameau through Schenker and Riemann as well as contemporary instructional manuals from Piston and the latest undergraduate harmony text to popular treatments such as those found in any guitar method book and Music Theory for Dummies.
2. Extensible: The tonal triad is capable of combinatorially generating other harmonic objects such as seventh chords, whether by adding sevenths or sixths, by triad superposition, or by third stacking. There are different opinions as to the correct analysis of the way this generation works, but the end result –– new objects that are harmonically similar to their progenitor –– significantly expands available harmonic material.
3. Matrix-defining: The triad's "shape" as a second-order maximally even structure connects it logically to the maximally even diatonic (I assume this recommends it based on our human aesthetic preference for symmetry, though I've never heard this argument specifically – only an amazement (which I share) at the triad's "fit" within a nested symmetry.) More importantly, beyond its symmetry and fit within the diatonic and other scales, the repetition of the triad shape at every level creates a defining coherence for the diatonic matrix. 
4. Aurally preferable (apart from any system or matrix): The major triad appears in nature in the lower partials of the harmonic series. This fact is often cited in conjunction with the questionable notion that, presented with the choice, humans have a physical or psychological preference for "natural" over "synthetic." (Unfortunately, to get at the essential minor triad in the harmonic series requires some intellectual juggling.) Another nature-preference argument comes from noting the relatively smaller (ergo simpler) frequency ratios of the tonal triad's constituent intervals, and relating this to humans' alleged preference (again, presented with the choice) for simple over more complex structures. Finally, there is a claim that a natural preference for the triad's sound per se is internal –– somehow wired into the human brain/psyche. Cognitive science has been enlisted to demonstrate this claim which, if it could be done, would lend credence to the "We all like it" argument, a statistical syllogism that derives first-person plural status from a sufficiently large sample of first-person singular preferences. The unacknowledged underbelly of this attempt to ally with science in order to get to we, is that it can be easily confused with the discredited, but often employed, rhetorical argumentum ad populum with a little ad baculum thrown in for spice. At any rate, any applicable valid science here continues to be surrounded by a lot of big ifs. As far as I know, cognitive science is continuing to tell us that the innate preference feature will be thoroughly understood by next Sunday. So stay tuned if you believe the outcome might justify your personal listening preferences or provide rocks to throw at composers who refuse to comply with nature.

This list resembles Euclid's axioms at least in the sense that each of the four basic categories (as proposition sets) is independent of the others. In particular (looking ahead), categories 1, 2 and 3 all offer pragmatic techne suggestions and useful game rules for composers of tonal works past and present and are easily seen to have nothing to do with the preference propositions alleged in category 4. The first three can stand untouched whether or not the triad object sounds good to you or me or anybody.

If this decoupling of sound from procedure is difficult to swallow, try this old philosophers' trick (usually done as an imaginary(?) conversation with the devil).

Imagine an extraterrestrial visiting Earth who, when encountering chords as simultaneities,  experiences pleasure from those intervals found consecutively in the higher overtones (the higher the overtones, the more pleasant the sensation) and excruciating pain from intervals appearing in the lower partials. Our ET's hearing is so sensitive that she can clearly distinguish overtones well over the Pythagorean comma, creating a harmonic preference that is very odd to us: the higher the partials, the closer consecutive intervals get to unison; so she loves the near-unison, but the perfect octave down at the bottom is almost unbearable to her. On her home planet they also have consonant triads as verticals, but each triad's constituent intervals are so close together we Earthlings can only hear them as a single fuzzy tone. Well, this is unusual, but at least we can relate in that some of our own musicians and theorists have been investigating microtones for a long time, albeit not this radical an upside-down harmony preference. But then it gets really weird. She tells us that their melodies are generally stepwise with occasional leaps for effect and to avoid boredom; except that by "step" she means intervals from the lowest partials and by "leap" she means intervals toward the higher end. To illustrate she takes out something she calls a jPod and plays a recording of an old accompanied folk melody from her planet. To our ears it is a random jumble of sounds jumping all over the acoustic spectrum, but she smiles as it plays. We ask her to please turn it off. Her planet's way of forming "simple," enjoyable harmonic and melodic material is precisely the opposite of ours. We make one more try to understand and ask her to explain how her concepts of melody relate to harmony. She produces what she calls a jPad and we scroll through a document she tells us was written by an ancient philosopher-composer from her world named I. I. Fux simply titled Counterline. At first it makes no sense. Then gradually we realize that if we carefully switch certain words around, step <––> leap, consonant <––> dissonant, and a few more ––– and then if we re-read the teacher-student conversation, leaving the "rules" exactly as they are, just switching a few basic definitions ....... hmmm.... Our mind wanders out of music and into math for some reason ––– we vaguely remember something about duals, dual spaces, dual theorems, switching out points for lines .... hmmmm.  Our reverie is disturbed by an obnoxious sound like the rapid repetiton of the highest and lowest notes on a piano. It's our visitor's jPhone. She says she must return immediately. Her planet's North Polar Cap has declared war on the South Polar Cap again. Some things, beside the laws of physics, are the same across the universe. We ask her to accept a musical gift to remember us by –– an accordion. She politely refuses. We understand, of course, and wish her well. She steps into the old abandoned phone booth and disappears. Down on the ground we see the jPod she must have accidentally dropped. Hmmmmmmm.......

________________________________

[1] Arnold Schoenberg (in Schoenberg, ed. Merle Armitage. 1937. p. 267)

[2] It is my understanding that "music analysis" conceived as an independent discipline (and today considered as all but synonymous with "music theory") began in earnest only a couple centuries ago. This makes sense since music analysis is dependent on a sufficiently large body of artifacts that somehow managed magically to appear (as well as to be shared and enjoyed) without any independently coherent analytical theory to speak of. It took some time for these artifacts to accumulate before the professional analyst could make an appearance. This answers how (academy oriented) analysis became possible, but fails to address the question of why it was, and still is, considered indispensable. Or often why it's helpful at all. (And a host of other questions.) C.H. Langford, writing about G.E. Moore's "analytical paradox," sums up my own conundrum regarding analysis:

Let us call what is to be analysed the analysandum, and let us call that which does the analysing the analysands. The analysis then states an appropriate relation  of equivalence between the analysandum and the analysands. And the paradox of analysis is to the effect that, if the verbal [music] expression representing the analysandum has the same meaning as the verbal [graphic, verbal] expression representing the analysands, the analysis states a bare identity and is trivial; but if the two ... expressions do not have the same meaning, the analysis is incorrect. (Langford, "The Notion of Analysis in Moore's Philosophy" in The Philosophy of G.E. Moore, ed. P.A. Schlipp, p.323)

But the paradox didn't stop Moore from philosophizing. None of this is meant to say that I don't see a place for analytical work to inform theory and composition –– that would be absurd, even to me. If I were asked how I can then even make a distinction, I would be forced to admit that I see theory as the attempt to show how things might be (while knowing that all possible paths will not all be chosen) and analysis as the attempt to show how things are (with no attempt to provide a normative framework for either composer or listener).  Still it's in my nature to rebel against the latter, even (or especially) when I find myself faking that pose in a group portrait.

Desperately Seeking Relevance: Music Theory Today [4]

"Modes of Imagining"
A few thoughts on rules, definitions, common notions,
requests, postulates, axioms, hypotheses,
and other stuff.

I will not run through all the modern axioms laid down by Russian boys on the subject, which are all absolutely derived from European hypotheses; because what is a hypothesis there immediately becomes an axiom for a Russian boy, and that is true not only of boys but perhaps of their professors as well, since Russian professors today are quite often the same Russian boys. And therefore I will avoid all hypotheses. – Ivan Karamazov[1]

I often think of music as a game. There may be a large, even infinite, number of ways to play, but a limited, relatively small set of rules defines any game. Over a long period of time, since originality is often an important goal, some of the rules might be altered so that boredom doesn't set in (possibilities tend to get used up as the history of play is extended). But the question then arises, which rules are you allowed to change and how much can you alter them before you find you are playing a different game altogether?

It doesn't take long before tic-tac-toe becomes boring. So you change the rule that says it has to be played on a two-dimensional surface. Is tic-tac-toe in three dimensions still tic-tac-toe? I think most people would say yes. But let's say you stick to two dimensions, expand the grid, change a few other rules, expand the grid some more and so on. Over the years, possibly centuries if you live so long, you come to realize you're playing a game that looks an awful lot like a game in another culture called Go. But are you still playing tic-tac-toe? Does it matter?

If we ask if Wagner was playing the same game as Josquin, it seems the answer would be both no and yes, depending on which rules you look at. Some of the rules in Wagner's music game would be unrecognizable to Josquin, and some of Josquin's would be anathema to Wagner. Others (very deep ones, I think) would be unchanged between the two – if not, we couldn't say that both composers were composing music that sounds enough alike such that no one is surprised when we call them both music.

The game metaphor suggests a few related knotty issues as well. When is a rule change called for? Is it simply a matter of avoiding boredom as I suggested at first, or are there other compelling reasons for rule changes? This raises issues of authority. Who decides a rule change is called for and what rule or rules to change? The composer? The performer? Who gets to decide whether or not to accept a rule change? The audience? The critic? History?

Now I suppose I could continue in this vein to segue into a critique of musical developments and experiments in the early-to-mid 20th century. It now seems to many, not without reason, that rule-changing itself became the game in those years. I can't deny this happened, but neither can I say that many of these rule changes were not desirable or necessary. The old game was getting awfully stale and a little too easy. But that's not the turn I want this metaphor to take. At least not yet. Instead, I'd like to move away from music a bit to see how players in another game have dealt with radical rule changes.

"In the nineteenth century, geometry, like most academic disciplines, went through a period of growth verging on cataclysm."[2]  (One might say that in the twentieth century, music, like most other arts, went through a period of cataclysm verging on growth.)

While there was an explosion of activity in many branches of mathematics, I think most would agree that a main cause for all the foment was a problem that kept bubbling up from just below the surface for centuries: the Fifth Postulate in Euclid's Elements. But the "parallel postulate" is not the issue here. The issue I'd like to dwell on is the word postulate per se, and how mathematics as a discipline learned to handle the problem of its "primitive concepts."

I just recently discovered that the Greek word Euclid used, which is generally translated as "postulates," was αιτεματα [aitemata], more accurately rendered as "requests." This is significantly different from the old, and still common, sense of postulates and axioms as self-evident propositions. It would seem that Euclid's Elements does not really begin with five postulates, nor with five axioms. Euclidean geometry begins with five requests. So then is the reader free to accept or reject any of these requests? Yes. But the consequence of rejecting any one of them is that Euclidean geometry doesn't work. – Well, that's not entirely true. As it turns out, the Fifth Request, concerning "parallel lines," can be left out, leaving much of the Euclidean edifice still standing, but the space it is left standing in need not be "flat" as common sense would dictate. And if the Euclidean Fifth Request is denied such that space is no longer "flat," some very strange objects and relations begin to appear – logically valid but playing havoc with our human sense of the way things are and the way they ought to be.

[N]o mathematician would invent something new in mathematics just to flatter the masses.... He who really uses his brain for thinking can only be possessed of one desire: to resolve his task. He cannot let external conditions exert influence upon the results of his thinking.... An idea is born; it must be moulded, formulated, developed, elaborated, carried through and pursued to its very end.[4]

[V]ery little of mathematics is useful practically, and ... that little is comparatively dull. The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences.[5] 

Still, it was not always so. Mathematicians are human, and tend to cling to their inherited realities like anyone else. It took centuries (and the development of a more safely liberal culture) for them to face up to the problem fully. In 1733, Giovanni Saccheri, having proven many theorems in hyperbolic geometry, dismissed his own work simply because it contradicted Euclid. To this day, outside the world of mathematics and science generally, the common sense (consensus: literally "feeling together") tells us that everything in Euclidean geometry is obviously true. Imagining the world otherwise, even "in theory," is nearly impossible. Outside of science fiction (NB!), suggesting a world where parallel lines meet is an abomination. To explain what our collective sense tells us, therefore, we accept Euclid's request because we need it to prove (justify) what we already know is true. We knowingly commit to this invalid argument because we need to get along in a world we can understand. "We do, doodley do, ... what we must, muddily must...." That's all. Is this beginning to sound familiar??

Despite being locked in by collective wisdom, in the nineteenth century mathematics found a way to stop begging this ancient question. Nikolai Lobachevsky devised an alternative system of geometry based on the negation of the Fifth Postulate (we may as well return to this commonly accepted translation now, my point having been made (I think)). Lobachevsky called the geometry he built an "imaginary" geometry.[6]  Around the same time, Janos Bolyai simply deleted the Fifth Postulate and termed what could then be deduced from the definitions and the first four postulates alone the "absolute geometry."[7] In 1854, Bernhard Riemann (who Milton Babbitt liked to call "the good Riemann") built a spherical geometry where there are no parallel lines (paving the way to the general theory of relativity). There were now two non-Euclidean geometries. Mathematicians had finally freed themselves from trying to make the worlds of reality and imagination conform to the Euclidian sacred text, and they did this essentially by populating the world with other geometries – not to improve upon, and certainly not to take the place of Euclid, but to join him. Euclidean geometry became one among many valid geometries, and this caused a different problem. With more geometries appearing on the scene, what, if anything, was the connection between them?

Enter Felix Klein and his Erlangen Program of 1872 which was a synthesis of many of the then-existing geometries (including Cartesian, projective, and others – the exception was spherical) as models of the same "abstract geometry."[8] Remember that term.

Neither reverence for the past nor the common sense of reality prevailed. Euclid remained, but the Euclidean lock was broken.

I'm going to break off this ultima Thule of a digression here because a musical Erlangen Program is precisely the place where I would like to resume my commentary on music theory today (MTT) with a challenge that will take two more posts. In my next post I'll review a list of what I understand to be MTT's basic, mostly unexamined assumptions. These assumptions I see as tracking the early history of geometry described above. Full of potential, MTT has stopped short of Lobachevsky & company and settled into the safe holding mode of a brilliant but frightened Saccheri. Just what the Erlangen Program is in math and what it might suggest to music theory should then become clear with what I intend to be my final post in this thread. But I'm more than willing to extend it if the challenge is taken up in a meaningful way.

The recognition of frontiers implies the possibility of crossing them. It is just as urgent for musical theory to reflect on its own procedures as it is for music itself. It is the bitter fate of any theory worthy of the name that it is able to think beyond its own limitations, to reach further than the end of its nose. To do this is almost the distinguishing mark of authentic thinking.

––Theodor Adorno (Quasi una Fantasia)

___________________________

[1] Fyodor Dostoevskii. The Brothers Karamazov, Part 2, Book 5, Chapter 3. Tr. Richard Pevear and Larissa Volokhonsky. San Francisco: North Point Press, 1990. p. 235.

[2][3][6][7] ("Nineteenth Century Geometry," Stanford Encyclopedia of Philosophy)

[4] Arnold Schoenberg. "New Music, Outmoded Music, Style and Idea" (1946) (In Style and Idea)
[5] G.H. Hardy. A Mathematician's Apology (p.89)
[8] Felix Klein. "A Comparative Review of Recent Researches in Geometry."